Question: The following data points are the yearly salaries (in thousands of dollars) of the $4$ high school cheerleading coaches in Dakota County. $41,46,52,49$ Find the standard deviation of the data set. Round your answer to the nearest hundredth.
Solution: How to calculate standard deviation The formula for standard deviation (SD) is $\text{SD} = \sqrt{\dfrac{\sum\limits_^{{\lvert x-\bar{x}\rvert^2}}}{n}}$ where $\sum$ means "sum of", $x$ is a value in the data set, $\bar{x}$ is the mean of the data set, and $n$ is the number of values in the data set. Step 1: Finding ${\bar{x}}$ in $\sqrt{\dfrac{\sum\limits_^{{\lvert x-{\bar{x}}\rvert^2}}}{n}}$ $\bar{x}$ is the mean of the data set. See if you can find $\bar{x}$ yourself. If you're stuck, click "Explain" below the problem. Fill in the blank. $\bar{x} = $ Step 2: Finding ${\lvert x - \bar{x} \rvert^2}$ in $\sqrt{\dfrac{\sum\limits_^{{{\lvert x-\bar{x}}\rvert^2}}}{n}}$ This is the square of the distance from each data point $x$ to the mean (the deviation). For example, in the first line of the table below $x = 41$ is $\lvert 41 - {47} \rvert ={6}$ away from the mean, and ${6}^2 = 36$. Complete the table below. Data point $x$ Distance from the mean squared $\lvert x - \bar{x} \rvert^2$ $ 41$ Step 3: Finding ${\sum\lvert x - \bar{x} \rvert^2}$ in $\sqrt{\dfrac{{\sum\limits_^{{\lvert x-\bar{x}}\rvert^2}}}{n}}$ This is the sum $\sum$ of the square of the distances from each of the data points $x$ to the mean $\bar{x}$ (the deviations). Fill in the blank. $\sum\lvert x - \bar{x} \rvert^2 = $ Step 4: Finding ${\dfrac{\sum\lvert x - \bar{x} \rvert^2}{n}}$ in $\sqrt{{\dfrac{\sum\limits_^{{\lvert x-\bar{x}}\rvert^2}}{n}}}$ This is the sum $\sum$ of the square of the distances from each of the data points $x$ to the mean $\bar{x}$ (the deviations), divided by the number of data points $n$. Fill in the blank. $\dfrac{\sum\lvert x - \bar{x} \rvert^2}{n} = $ Step 5: Finding the standard deviation $\sqrt{\dfrac{\sum\limits_^{{\lvert x-\bar{x}\rvert^2}}}{n}}$ This is the square root of the sum $\sum$ of the square of the distances from each of the data points $x$ to the mean $\bar{x}$ (the deviations), divided by the number of data points $n$. This is it! This is the standard deviation! Fill in the blank. Round your answer to the nearest hundredth. $\text{SD} = \sqrt{\dfrac{\sum\limits_^{{\lvert x-\bar{x}\rvert^2}}}{n}} \approx $ The answer The standard deviation is approximately $4.06$ thousand dollars. [I want to see all of the steps at once]